The largest area you can enclose with your budget of $40 is 50 square feet.
Define Variables:
- Let x be the length of the east and west sides (in feet)
- Let y be the length of the north and south sides (in feet)
- Let A be the area of the vegetable patch (in square feet)
Express Costs:
Cost of east and west sides: 2x * $2/foot = 4x dollars
Cost of north and south sides: 2y * $1/foot = 2y dollars
Budget Constraint:
The total cost of the fencing cannot exceed the budget of $40.
Therefore, we have the equation: 4x + 2y = 40
Maximize Area:
We want to maximize the area A of the vegetable patch.
Area = length * width = xy
Solve for x and y:
From equation 3, we can express x in terms of y: x = (40 - 2y) / 4 = 10 - 0.5y
Substitute this expression for x in the area equation:
A = y * (10 - 0.5y) = 10y - 0.5y^2
Find Maximum Area:
To find the maximum area, we need to find the maximum value of the function A(y) = 10y - 0.5y^2.
This is achieved when the derivative A'(y) = 0:
A'(y) = 10 - 1y = 0
Solving for y, we get y = 10.
Calculate Maximum Area:
Substitute y = 10 back into the area equation:
A = 10 * 10 - 0.5 * 10^2 = 100 - 50 = 50 square feet
Therefore, the largest area you can enclose with your budget of $40 is 50 square feet.